At KS3 pupils have a chance to study 20th-century topics beyond the narrow sphere of international relations, so you can build in work on world health, environmental issues, refugees or the world body's peacekeeping role. Use pupils' research for a class debate set in various countries or historical junctures. The class could discuss whether it is worth joining the UN or continuing to pay towards it. "Witnesses" could use the UN's track record to fight their corner. At GCSE, pupils are most likely to look at the UN's role in the Cold War. The classic case studies are Korea in 1950 and Cuba in 1962.
Pupils could make timelines of each, perhaps with a limit of 10 events, then decide which of these events should involve the UN.